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Sean Monahan and Mikael Backlund also tallied for the Flames, who improved to 9-3 in the extra session this season. Karri Ramo stood tall with 35 saves.
"We just find a way and we believe in each other and we try to get as many pucks on the net as we can," Monahan said about his team pulling together after it was learned on Monday that defenseman Mark Giordano was lost for the season thanks to a torn bicep.
Sean Couturier and Mark Streit scored for the Flyers, who have dropped three of their last four. Steve Mason stopped 23 pucks in defeat.
Philly defenseman Nick Schultz appeared to have netted the game-tying goal at 4:58 of the third period, as he crashed the net and his own rebound deflected off his knee and past Ramo.
Despite a good goal call on the ice, a brief conference between officials somehow determined that the play could not be reviewed since Schultz pushed Ramo into the net. The score was disallowed.
"Called it a goal originally on the ice, but they huddled because ... I don't know," said Flyers head coach Craig Berube. "A linesman might have said something different, and they huddled. In their huddle they decided that there was incidental contact with the goalie."
The Flyers snapped to life and tied the game at 8:15 when Streit got a piece of a Michael Raffl shot from between the circles and flipped the rebound over Ramo's glove from in close.
In overtime, Johnny Gaudreau fed to a streaking T.J. Brodie up the left wing, and his pass into the slot hit Hudler's stick as he was marked by host defenseman Michael Del Zotto and trickled through Mason's pads to end the contest.
"I was down and it was a quick pass out and bang bang play, hit my pad and kept going through,"
Monahan ripped one home from the right wing 6 1/2 minutes into the contest thanks to a pass from Mason Raymond after Hudler stripped Del Zotto in the neutral zone leading to Calgary's transition.
The Flames went up 2-0 as Lance Bouma circled behind Mason and slipped a pass across the crease for Backlund's easy tap-in just before the five-minute mark of the second.
Ramo made two consecutive sprawling stops on hard shots by Jakub Voracek on a Philly power play inside of 20 seconds left in the middle frame to keep it a two-goal bulge.
"Ramo is the story of our past two weeks. He is our best player out there. He is giving us lots of confidence; he is giving us key saves at the right time," noted Flames head coach Bob Hartley.
Seconds after that carry-over advantage expired, the hosts got on the board as Couturier banged home a cross-crease pass from Ryan White at 53 seconds of the third.
Notes: Monahan's goal was his 22nd of the season, matching his personal best set last season when he was a rookie ... Calgary recorded 33 blocked shots ... Flyers captain Claude Giroux recorded his 300th career assist on Streit's goal, making him one of 13 players in franchise history to reach the mark. Giroux stand three helpers away from tying Eric Desjardins for 12th place.